3.19 \(\int \frac {\sin ^{-1}(a x)^2}{x^3} \, dx\)

Optimal. Leaf size=44 \[ -\frac {a \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{x}+a^2 \log (x)-\frac {\sin ^{-1}(a x)^2}{2 x^2} \]

[Out]

-1/2*arcsin(a*x)^2/x^2+a^2*ln(x)-a*arcsin(a*x)*(-a^2*x^2+1)^(1/2)/x

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Rubi [A]  time = 0.08, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4627, 4681, 29} \[ -\frac {a \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{x}+a^2 \log (x)-\frac {\sin ^{-1}(a x)^2}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[ArcSin[a*x]^2/x^3,x]

[Out]

-((a*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/x) - ArcSin[a*x]^2/(2*x^2) + a^2*Log[x]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 4627

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcSi
n[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1))/Sqrt[1
- c^2*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 4681

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(
(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n)/(d*f*(m + 1)), x] - Dist[(b*c*n*d^IntPart[p]*(d + e*x
^2)^FracPart[p])/(f*(m + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSi
n[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p
+ 3, 0] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {\sin ^{-1}(a x)^2}{x^3} \, dx &=-\frac {\sin ^{-1}(a x)^2}{2 x^2}+a \int \frac {\sin ^{-1}(a x)}{x^2 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {a \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{x}-\frac {\sin ^{-1}(a x)^2}{2 x^2}+a^2 \int \frac {1}{x} \, dx\\ &=-\frac {a \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{x}-\frac {\sin ^{-1}(a x)^2}{2 x^2}+a^2 \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 44, normalized size = 1.00 \[ -\frac {a \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{x}+a^2 \log (x)-\frac {\sin ^{-1}(a x)^2}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSin[a*x]^2/x^3,x]

[Out]

-((a*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/x) - ArcSin[a*x]^2/(2*x^2) + a^2*Log[x]

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fricas [A]  time = 1.31, size = 44, normalized size = 1.00 \[ \frac {2 \, a^{2} x^{2} \log \relax (x) - 2 \, \sqrt {-a^{2} x^{2} + 1} a x \arcsin \left (a x\right ) - \arcsin \left (a x\right )^{2}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)^2/x^3,x, algorithm="fricas")

[Out]

1/2*(2*a^2*x^2*log(x) - 2*sqrt(-a^2*x^2 + 1)*a*x*arcsin(a*x) - arcsin(a*x)^2)/x^2

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giac [B]  time = 0.18, size = 82, normalized size = 1.86 \[ \frac {1}{2} \, {\left ({\left (\frac {a^{4} x}{{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} {\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{x {\left | a \right |}}\right )} \arcsin \left (a x\right ) + 2 \, a \log \left ({\left | x \right |}\right )\right )} a - \frac {\arcsin \left (a x\right )^{2}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)^2/x^3,x, algorithm="giac")

[Out]

1/2*((a^4*x/((sqrt(-a^2*x^2 + 1)*abs(a) + a)*abs(a)) - (sqrt(-a^2*x^2 + 1)*abs(a) + a)/(x*abs(a)))*arcsin(a*x)
+ 2*a*log(abs(x)))*a - 1/2*arcsin(a*x)^2/x^2

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maple [A]  time = 0.04, size = 43, normalized size = 0.98 \[ -\frac {\arcsin \left (a x \right )^{2}}{2 x^{2}}-\frac {a \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{x}+a^{2} \ln \left (a x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsin(a*x)^2/x^3,x)

[Out]

-1/2*arcsin(a*x)^2/x^2-a*arcsin(a*x)*(-a^2*x^2+1)^(1/2)/x+a^2*ln(a*x)

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maxima [A]  time = 0.53, size = 40, normalized size = 0.91 \[ a^{2} \log \relax (x) - \frac {\sqrt {-a^{2} x^{2} + 1} a \arcsin \left (a x\right )}{x} - \frac {\arcsin \left (a x\right )^{2}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)^2/x^3,x, algorithm="maxima")

[Out]

a^2*log(x) - sqrt(-a^2*x^2 + 1)*a*arcsin(a*x)/x - 1/2*arcsin(a*x)^2/x^2

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\mathrm {asin}\left (a\,x\right )}^2}{x^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asin(a*x)^2/x^3,x)

[Out]

int(asin(a*x)^2/x^3, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {asin}^{2}{\left (a x \right )}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asin(a*x)**2/x**3,x)

[Out]

Integral(asin(a*x)**2/x**3, x)

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